Fill in the following tables with eight bit numbers Discuss

Fill in the following tables with eight bit numbers. Discuss when overflow/underflow occurs addition of two\'s complement. Show an example of overflow for the addition of two, 4-bit numbers in two\'s complement. Show an example of underflow for the addition of two, 4-bit numbers in two\'s complement. Consider a function that takes as input two 2-bit numbers and produces as output a 3-bit sum. Write the truth table for this function. Consider one 2-bit number to be represented by A_1 A_0 and the other by B_1 B_0 using 2-bit binary (base 2) positional numbers. For example, A_1 A_0 = 10 means that the corresponding number is 2. The 3-bit output is represented by XYZ using binary (base 2) positional numbers. For example, XYZ = 110 means the output is 6.

Solution

3 a)

3 b)

3 c)

Overflow rule : If two numbers with the same sign (both positive or both negative) are added, then overflow occurs if and only if the result has the opposite sign.

Overflow occurs when the number that you trying to represent is out of the range of numbers that can be represented. In your example you are using 4-bits two\'s complement, that means you can represent any number in the range -8 (1000) up to +7 (0111). The result of your subtraction 2-1 is +1, a number that lies within the range of representation.

When we add a negative and a positive operand, the result will always be in the range of representation. Overflows occur when we add two numbers with the same sign (both positive or both negative) and the result has the opposite sign.

Most of misunderstanding surrounding carry-out and overflow comes from the fact we use the carry-out as one of the parameters to generate overflow flag. They are strongly related but they are not the same thing.

When adding numbers in two\'s complement, if the carry-out and the carry-on into the most significant bit (sign bit) are different that means an overflow has occurred.

Let\'s see two negative operands with a positive result:

The carry-out is 1 and the carry-on to sign bit (MSB) is 0.

And now, an example of two positive operands with a negative result.

Underflows refer to floating point underflow, where an operation result in a number that is too small to be representable. For example, if the exponent part can represent values from 127127 to 127127, then any number with absolute value less than 21272127 may cause underflow.

If you go with two\'s complement addition, then it is as you say: carry of 1 is underflow.

4. Truth table for function which accepts two 2-bit numbers and returns one 3-bit sum

first operand = 2 binary - 10
second operand = 3 binary - 11
sum = 5 binary - 101

Hence, output is 101 i.e. 5

	Decimal	Signed Magnitude	1s Complement	2s complement
	-67	1	0111100	0111101
+	42	0	0010101	0010110
	-25	1	0000110	0000111